This semester we’ll have talks on two themes i. Tropical curves and their divisor theory, ii. Green’s conjecture on syzygies of canonical curves. We’ll start with the basics of tropical part (about 2-3 weeks) and then move to basics of Green’s conjecture part (2-3 weeks). We’ll then move on to more advanced aspects of both these topics.

Update from 8-th September meeting:

This semester (probably the next) we plan to focus on aspects of syzygies: combinatorial, algebraic, geometric. We’ll devote the first few lectures to the basics of syzygies.

In the future, we plan to cover various aspects of the interplay between Combinatorics and Commutative Algebra-Algebraic Geometry. Some possible topics are

We plan to pick a theme every semester followed by talks by local faculty and students. These talks will be introductory and will assume very little background. We will then invite speakers from outside for more specialized talks on this theme.

January 8, 2017: I will spend time at MFO, Schroedinger Institute in Vienna and IHES Paris this year.

November 4, 2016: Back to London and expect to be here till the end of December.

October 24, 2016: Visiting Rice University, 26, 27 October and then Minneapolis for the AMS Special Session on “Chip Firing, Divisors on Graph and Simplicial Complex”, October 28-30. My talk is on Saturday, 29 October.

October 6, 2016: We are holding a Tropical Geometry Seminar at Imperial this term. Our first meeting will be at Huxley 109, 4pm tomorrow (October 7). Please drop by or write to me if you want to know more about it.

September 13, 2016: Back to London and expect to be in this area throughout September.

Tropical geometry and its applications indicate a theory of syzygies over polytope semirings. The broad goal of this project is to developed this notion and study applications, for instance to canonical embeddings of tropical curves (as developed by Haase, Musiker and Yu) along the lines of Green’s study of syzygies of canonical algebra curves. I recently completed a paper that takes first steps in this direction.

Tropical Graph Curves:

The broad goal in this project is to investigate the construction of “good” tropicalizations of canonical algebraic curves. More precisely, we mean tropicalizations that provide valuable information about the Berkovich analytification of the algebraic curve. This led to the concept of tropical graph curves. Using this we construct tropicalizations that capture the topology of the Berkovich analytification for algebraic curves whose graph underlying the Berkovich analytification is a three-connected planar graph. See below for preprints.

Combinatorial Brill-Noether for Graphs via Commutative Algebra:

We are investigating the Brill-Noether theory for Graphs a la Baker and Norine via commutative algebra. It turns out that the Brill-Noether theory on graphs is closely related to Boij-Seoderberg style problems on quotient rings of certain binomial ideals associated to the graph. We are currently investigating the free resolution of the residue field over these quotient rings. For instance, classifying graphs for which the ring is Golod. This is work in progress with A. Fink.

Commutative Algebra of Generalised Frobenius Numbers:

One goal of this project is to develop commutative algebraic interpretations of generalised Frobenius numbers. The combinatorial aspects have been developed in the work of Beck and Robins and the recent work of Aliev, De Loera and Louveaux. This joint work with B. Smith.

In January 2016, I taught an LTCC course on Tropical Geometry. I previously taught introduction to Abstract Algebra, Math 113 in Berkeley. Previously, I have taught courses at both undergraduate and graduate levels. At the undergraduate level I have taught courses on linear and discrete mathematics and calculus at Georgia Tech and the graduate level a course on combinatorial commutative algebra at Saarland University.

As a postdoc, I have particularly enjoyed working with graduate
students and undergraduate students: mentoring, sharing my experiences with them and learning from them has been great fun! I look forward to working more with undergraduate and graduate students in the future.

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